Theorem abs2sqlei

Description: The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007)

	Ref	Expression
Hypotheses	abs2sqlei.1	⊢ 𝐴 ∈ ℂ
	abs2sqlei.2	⊢ 𝐵 ∈ ℂ
Assertion	abs2sqlei	⊢ ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) )

Proof

Step	Hyp	Ref	Expression
1		abs2sqlei.1	⊢ 𝐴 ∈ ℂ
2		abs2sqlei.2	⊢ 𝐵 ∈ ℂ
3	1	absge0i	⊢ 0 ≤ ( abs ‘ 𝐴 )
4	2	absge0i	⊢ 0 ≤ ( abs ‘ 𝐵 )
5	1	abscli	⊢ ( abs ‘ 𝐴 ) ∈ ℝ
6	2	abscli	⊢ ( abs ‘ 𝐵 ) ∈ ℝ
7	5 6	le2sqi	⊢ ( ( 0 ≤ ( abs ‘ 𝐴 ) ∧ 0 ≤ ( abs ‘ 𝐵 ) ) → ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) ) )
8	3 4 7	mp2an	⊢ ( ( abs ‘ 𝐴 ) ≤ ( abs ‘ 𝐵 ) ↔ ( ( abs ‘ 𝐴 ) ↑ 2 ) ≤ ( ( abs ‘ 𝐵 ) ↑ 2 ) )